Honestly, as someone who has an interest working with large data-sets, I took her remark ("who wants to work at a bank?") as somewhat insulting. This specific interviewer was not a hiring manager, but rather questioning me for "cultural fit". I asked how long she had been with said company and she admitted "less than three months," which surprised me. I really got the sense that her hating on banks was regurgitated water-cooler verbatim, because she definitely had no experience working for one.
As a young professional working in technology, I think banks present interesting big-data challenges.
That said, I work with other software developers who have worked for a few of the major banks and they said it was awful, basically, I don't think its bank-specific, but there is a major divide between a company that's primary function is to produce software and one whose isn't, and banks do not exist to produce software, its just a means to and end for them, so projects are managed as such and developers are treated as such.
And you've just pretty much guaranteed that you'll get an offer for no more than $Y.
I wouldn't have mentioned who the other offer came from. It's not something they need to know.
Are you actually flexible? Surely there's an amount you like. I would have said something like, "I have an offer for $X a year, but I'd much rather prefer working for you, so my minimum would be $Y," where $Y < $X.